T Matrix

From:"Ludden, Thomas" Subject:[NMusers] T Matrix Date:Thu, March 7, 2002 2:02 pm Dear nmusers, The following is provided by Dr. Stuart Beal in response to the inquiry about the T matrix. Tom Ludden _________________________________________________________________ _________________________________________________________________ On occasion, people want a little more information about the T matrix that sometimes appears in output from the Covariance Step. Here is an updated help item from the current NONMEM Version VI (under development). It notes that the T matrix can be output when either the R or S matrix is singular. In such a case, in order to obtain information about just where the singularity is located, one should examine the R or S matrix itself. Stuart Beal ------------------------------------------------------------------- | | | COVARIANCE MATRIX OF ESTIMATE | | | ------------------------------------------------------------------ MEANING: NONMEM's estimate of the precision of its parameter estimates CONTEXT: NONMEM output DISCUSSION: From asymptotic statistical theory, the distribution of the parameter estimates is multivariate normal, with a variance-covariance matrix that can be estimated from the data. NONMEM supplies such an estimate of the variance-covariance matrix. This matrix is not to be confused with either SIGMA, the covariance matrix for the second level random effects, or with OMEGA, the covariance matrix for the first level ran- dom effects. These two matrices estimate the variability of epsilons or etas, respectively, about their means. The variance-covariance matrix of the parameter estimates, on the other hand, measures the variability under the assumed model of the parameter estimates across (imagined) replicated data sets, using the design of the real data set. The following is an example of the NONMEM output giving the estimate of the variance-covariance matrix. **************** COVARIANCE MATRIX OF ESTIMATE ******************** TH 1 TH 2 OM11 OM12 OM22 SG11 TH 1 3.94E+01 TH 2 -6.89E+00 3.67E+02 OM11 -4.31E-02 3.17E-02 -2.92E-04 OM12 ......... ......... ......... ......... OM22 8.65E-02 -5.05E-01 2.71E-04 ......... 1.26E-02 SG11 -1.01E-02 -1.85E-02 -2.11E-05 ......... -3.10E-04 3.10E-05 The matrix (which is symmetric) is given in lower triangular form. In this example, the 2x2 matrix, OMEGA, was constrained to be diagonal; the omitted entries above (.........) indicate that OM12 is not estimated, and consequently has no corresponding row/column in the variance-covariance matrix. When the size of the array exceeds 75x75, a compressed form is printed in which the omitted entries (.........) are not printed. The compressed form may also be requested for arrays smaller than 75x75 (See $covariance). The (estimated) variance-covariance matrix is computed from the R and S matrices; it is Rinv*S*Rinv, where Rinv is the inverse of the R matrix. The R matrix is the Hessian matrix of the objective function, evaluated at the parameter estimates. The S matrix is obtained by adding the cross-product gradient vectors of the objective function, evaluated at the parameter estimates, across the individual records of the data set. The inverse variance-covariance matrix R*Sinv*R is also output (labeled as the Inverse Covariance Matrix), where Sinv is the inverse of the S matrix. This matrix can be used to develop a joint confidence region for the complete set of population parameters. As we usually develop a confidence region for a very limited set of population parameters, this use of the inverse variance-covariance matrix is somewhat limited. An error message from the Covariance Step stating that the R matrix is not positive semidefinite indicates that the parameter estimates do not correspond to a true (local) minimum and are not to be trusted. An error message stating that the R matrix is positive semidefinite, but singular, indicates that the objective function is flat in a neighborhood of the parameter estimate, and so the minimum is not really unique, and there is probably some overparameterization. An error message stating that the S matrix is singular indicates strong overparameterization. When the S matrix is judged to be singular, but the R matrix is posi- tive definite, the T matrix, R*Spinv*R, where Spinv is a pseudo- inverse of the S matrix, is output. Just as with the inverse variance-covariance matrix, T can be used to develop a joint confi- dence region for the complete set of population parameters. When the R matrix is judged to be singular, but S is nonsingular, the T matrix, R*Sinv*R, is output. (This cannot be called the inverse covariance matrix, as the covariance matrix does not exist.) Just as with the inverse variance-covariance matrix, T can be used to develop a joint confidence region for the complete set of population parame- ters. There are options that allow the variance-covariance matrix to be com- puted as either Rinv or Sinv. Asymptotic statistical theory suggests that these matrices are appropriate under the additional assumption that the objective function is indeed additively proportional to minus twice the likelihood function for the data. (See standard error of estimate, correlation matrix of estimate). REFERENCES: Guide I, section C.3.5.2 (p. 20) REFERENCES: Guide II, section D.2.5 (p. 21) REFERENCES: Guide V, section 5.4 (p. 43), 13.3 (p. 145)